Name
ID
MATH 251
Sections 506
Exam 1
Spring 2006
P. Yasskin
Multiple Choice: (5 points each. No part credit.)
1-9
/45
10
/20
11
/15
12
/10
13
/15
Total
/105
1. A fly travels along the path ⃗
rt = 3 sin t, −4 sin t, 5 cos t.
Find the arc length traveled by the fly between 3, −4, 0 and 0, 0, −5.
a. π
b.
c.
d.
e.
2
π
3π
2
2π
5π
2
̂ to the curve ⃗rt = 3 sin t, −4 sin t, 5 cos t.
2. Find the unit binormal B
a.
b.
c.
d.
e.
3 , −4 , 0
5 5
3 , 4 ,0
5 5
4 , −3 , 0
5 5
4 , 3 ,0
5 5
−3 , 4 , 0
5 5
1
3. Find the vector projection of the vector ⃗
u = ⟨1, −2, 2⟩ onto the vector ⃗v = ⟨−1, 1, 1⟩.
a.
b.
c.
d.
e.
−1 , 1 ,
3 3
1 , −1 ,
3 3
1 , −1 ,
9 9
−1 , 2 ,
9 9
1 , −2 ,
9 9
1
3
−1
3
−1
9
−2
9
2
9
4. Find the plane parallel to 4x − 3y + 2z = 5 which passes through the point P = 1, 2, 3.
What is the z-intercept of this plane?
a. 2
b. 4
c. 6
d. 8
e. 10
5.
y
The plot at the right is the contour plot of which function?
a. The hyperbolic paraboloid z = x 2 − y 2
4
2
b. The hyperbolic paraboloid z = xy
c. The hyperboloid
z=
1 + x2 + y2
d. The cone
z=
x − 1 2 + y 2
e. The elliptic paraboloid
z = x 2 + y − 1 2
-4
-2
2
4
x
-2
-4
2
6. If r =
x 2 + y 2 , find
∂ 2 r 3, 4.
∂x∂y
a. 12
b.
c.
d.
e.
5
12
25
12
125
−12
125
−12
5
7. The graph of a function z = fx, y passes through the point 2, 4, 8 and has f x 2, 4 = −3 and
f y 2, 4 = 5. Use the linear approximation to approximate f2. 2, 3. 9
a. −1. 1
b. 1. 1
c. 6. 9
d. 7. 9
e. 9. 1
8. Duke Skywater is travelling through the galaxy. At the present time he is at the point with galactic
coordinates P = 40, 25, 53 (in lightyears), and his velocity is ⃗v = . 1, −. 2, . 3 (in lightyears /year).
He measures the polaron density to be U = 4300 polarons/cm 3 and its gradient to be ⃗
∇U = 3, 2, 1
dU
3
polarons/cm /lightyear. Find the rate
at which Duke sees the polaron density changing (in
dt
3
polarons/cm /year).
a. 0. 2
b. 1. 0
c. 223
d. 223. 2
e. 224
3
9. The density of carbon monoxide in a room is given by δ = z ln9 + x 2 − y 2 . If you start at the point
3, 4, 2, in what direction should you move to decrease the carbon monoxide density as fast as
possible?
a. −6, −8, − ln 2
b. −6, 8, − ln 2
c. 6, −8, − ln 2
d. −6, 8, ln 2
e. 6, −8, ln 2
Work Out: (Points indicated. Part credit possible. Show all work.)
10. Find the equation of the plane tangent to each of the following surfaces:
a.
10 points
z = fx, y = xe xy
b.
10 points
xe yz + ye xz = 5
at
at
x, y = 2, 0
x, y, z = 2, 3, 0
4
11.
12.
The radius of a cylinder is currently r = 5 cm and its height is h = 10 cm.
cm
cm
Its radius is increasing at dr = 0. 3
and its height is decreasing at dh = −0. 2
.
dt
dt
sec
sec
Is its volume increasing or decreasing and at what rate?
15 points
10 points
Find
∂z
∂v
If z = x 2 y 3 where x = xu, v and y = yu, v satisfy
∂x 3, 4 = 5
∂x 3, 4 = 6
x3, 4 = 2
∂u
∂v
∂y
∂y
y3, 4 = 1
3, 4 = 7
3, 4 = 8
∂u
∂v
u,v=3,4
.
5
13.
15 points
Find the point on the ellipsoid
f = −x − 2y + 4z is a minimum.
2
x 2 + y + z 2 = 3 at which the function
16
4
You may use any method but Lagrange multipliers is easier than eliminating a variable.
6